\[ \int \frac {a+b \text {ArcSin}(c+d x)}{(c e+d e x)^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{7 d e \left (e \left (d x +c \right )\right )^{\frac {7}{2}}}+\frac {12 b \EllipticE \left (\frac {\sqrt {-d x -c +1}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {e \left (d x +c \right )}}{35 d \,e^{5} \sqrt {d x +c}}-\frac {4 b \sqrt {1-\left (d x +c \right )^{2}}}{35 d \,e^{2} \left (e \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {12 b \sqrt {1-\left (d x +c \right )^{2}}}{35 d \,e^{4} \sqrt {e \left (d x +c \right )}} \]
command
integrate((a+b*arcsin(d*x+c))/(d*e*x+c*e)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (5 \, b d \arcsin \left (d x + c\right ) + 5 \, a d + 2 \, {\left (3 \, b d^{4} x^{3} + 9 \, b c d^{3} x^{2} + {\left (9 \, b c^{2} + b\right )} d^{2} x + {\left (3 \, b c^{3} + b c\right )} d\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}\right )} \sqrt {d x + c} e^{\frac {1}{2}} + 6 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \sqrt {-d^{3} e} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )\right )} e^{\left (-5\right )}}{35 \, {\left (d^{6} x^{4} + 4 \, c d^{5} x^{3} + 6 \, c^{2} d^{4} x^{2} + 4 \, c^{3} d^{3} x + c^{4} d^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {d e x + c e} {\left (b \arcsin \left (d x + c\right ) + a\right )}}{d^{5} e^{5} x^{5} + 5 \, c d^{4} e^{5} x^{4} + 10 \, c^{2} d^{3} e^{5} x^{3} + 10 \, c^{3} d^{2} e^{5} x^{2} + 5 \, c^{4} d e^{5} x + c^{5} e^{5}}, x\right ) \]